Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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2
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0answers
22 views

Gap between “fibration” and “fiber bundle”.

There are fibrations $E \rightarrow B$ which are not fiber bundles. Example: $E = [0,1]^2 / \text{middle vertical line segment}$ and $B=[0,1]$. In this example, $E$ has the homotopy type of a total ...
3
votes
1answer
46 views

Clarification about the Thom-Pontrjagin construction as explained in Bredon's book

In Bredon's book, at page 118-119, there is a little chapter about the Thom-Pontrjagin construction, and I'm trying to follow the reason depicted there. He starts with a map $f \colon R^{n+k}\to ...
4
votes
1answer
52 views

Showing that $\mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.

Intuitively, I can see why this is. I've found a few threads about this, but they only provide, for example, a deformation retraction of $I \times I$ to its diagonal $D = \{ (x,x) \in I \times I \}$, ...
2
votes
0answers
34 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
5
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1answer
80 views

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$. I need to show that a homotopy equivalence between them doesn't exist, but it seems like the homology groups of the spaces ...
7
votes
1answer
114 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...
4
votes
1answer
406 views

local homology group

The question is: $X\in \mathbb{R}^{n}$ is the subspace ${(x_{1},...,x_{n}\mid x_{n}\geq 0)}$, and let $Y$ is the subspace with $x_{n}=0$. let $x\in Y$, calculate the local homology of $X$ at $x$. ...
4
votes
1answer
43 views

isotopy equivalence between manifolds

The definition below is from Encyclopaedia of Mathematics: Volume 6. Question: For any $n\geq 1$, is the $n$-dimensional closed cube $$[0,1]^n=[0,1]\times [0,1]\times\cdots \times[0,1]$$ isotopy ...
4
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0answers
48 views

Composition, fibrations?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. Suppose that $f$ is a homotopy equivalence. My question is, does it follow that $f$ is a fiber ...
1
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1answer
36 views

Deleting a contractible subspace is the same as deleting a point

Let $X$ be a topological space and $A$ and $B$ are subspaces of $X$. Suppose that $A$ is contractible. I know that taking the quotient does not affect the homotopy type, that is $X/A$ is homotopy ...
0
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0answers
33 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
2
votes
3answers
117 views

$\mathbb{R}^{2}$ and $\mathbb{R} \times [0, +\infty]$ are homotopy equivalent, but not homeomorphic

So, let's consider $M=\mathbb{R}^{2}$ and $N= \mathbb{R} \times [0, +\infty]$ - two topological spaces. Since $\pi_{1}(M)=\pi_{1}(\mathbb{R}) \times \pi_{1} (\mathbb{R}) = \{0 \}$ (since $\mathbb{R}$ ...
1
vote
2answers
43 views

Pullbacks and homotopy equivalences

Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it ...
2
votes
0answers
91 views

Deformation retract of a triangle

Let $X \subset \mathbb{R}^2$ be a triangle equipped with the topology induced by the euclidean topology on $\mathbb{R}^2$ and let $Y \subset X$ be the subset made of two sides of the triangle. I need ...
0
votes
0answers
48 views

$\Bbb R^5$ without two circles and line

How to prove that $\Bbb R^5$ without two $S^1$ and one $\Bbb R^1$ is homotopiclly equivalent to wedge of $(\Bbb R^5\backslash \Bbb R^1)$ and two copies of $(\Bbb R^5\backslash S^1)$? The only idea ...
4
votes
1answer
50 views

Fiber bundle with null-homotopic fiber inclusion

It is an exercise from Hatcher (exercise 31, page 392): For a fiber bundle $F \to E \xrightarrow{p} B$ such that the inclusion $F \hookrightarrow E$ is homotopic to a constant map, show that the long ...
3
votes
0answers
44 views

Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?). What ...
0
votes
0answers
27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
1
vote
2answers
53 views

Action of the fundamental group

Suppose that $M$ is a smooth manifold. Is it true that the fundamental group $\pi_1(M)$ always acts on $M$? If so, how this action is defined? EDIT: Of course I want my action to be nontrivial, say ...
2
votes
0answers
89 views

Confusion with $CW$ approximation

Hatcher give the following defintion of $CW$ approximation(page 352): Given a pair $(X,A)$ where the subspace $A\subset X$ is a nonempty $CW$ complex, an n-connected CW model for $(X,A)$ is an ...
2
votes
1answer
47 views

Zero in the Grothedieck group of the derived category

I have a problem. I was wondering whether there is a precise answer to the following question. Let $\mathcal{A}$ be an abelian category and $\mathcal{D}^b(\mathcal{A})$ its bounded derived category. ...
2
votes
1answer
39 views

Higher homotopy groups of wedge of circles.

Using van-kampen theorem, Fundamental group of wedge of n-circles is free group on n-generator. But I don't know how to calculate higher homotopy groups of wedge of spaces, in particular circles. I ...
1
vote
0answers
42 views

special kinds of homotopies

Let $X$ and $Y$ be two homotopy-equivalent topological spaces. That is, there exists maps $f:X \to Y$ and $g:Y \to X$ such that $g \circ f \simeq 1_X$ and $f \circ g \simeq 1_Y$. $1_X$ and $1_Y$ are ...
4
votes
1answer
85 views

Fundamental group of open subsets of $\mathbb{R}^n$

Suppose that $U$ is an open subset of $\mathbb{R}^n$. What can be said about its fundamental group? I'm sure that the answer should be well known, since this is rather natural question.
1
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0answers
34 views

Show that there exist no retraction from $RP^n$ to $RP^k$ if n>k.

I am trying this problem from Hatcher's algebraic topology book(4.2.1). If r:X$\rightarrow$A is retraction then I know that this induces injective map in the fundamental group level through inclusion ...
2
votes
1answer
36 views

Equivalence between derived categories preserve distinguished triangles

I have a problem: Is it true that every equivalence between derived categories preserve their distinguished triangles? Thanks very much!
0
votes
0answers
25 views

how to calculate relative homotopy groups?

I am studying nth relative homotopy groups from Hather.For a pair (X,A) where A$\subset$X nth-relative homotopy groups is defined by homotopy class of maps$(I^n,\delta I^n,J^{n-1})$ ...
0
votes
2answers
55 views

Let $ \gamma $ be the unit circle then $ \int_\gamma \frac {dz}{z^2 − 2z} = -\pi i$

Definition: If $ f $ is holomorpic in $G$ and gamma $\gamma$ is $G$-homotophic to a point then gamma is G-contratible and if gamma is G-contractible then $ \int_\gamma f = 0 $. By splitting the ...
2
votes
0answers
31 views

definition of homology via spectra

Let $K(\mathbb{Z}, n)$ denote a Eilenberg-Mac Lane space, characterized by $H^n(X, \mathbb{Z})=[X, K(\mathbb{Z}, n)]$ for all spaces $X$. In stable homotopy theory, the corresponding homology theory, ...
1
vote
1answer
23 views

configuration-spaces and iterated loop-spaces

In the paper Configuration-Spaces and Iterated Loop-Spaces. Graeme Segal, page 213-214, it is obtained that the labelled configuration space $C_n$ is homotopy equivalent to a topological monoid ...
8
votes
2answers
406 views

Very basic homotopy question

I'm brand new to homotopy theory so I'm sure this question is utterly stupid. But anyway I'm trying to understand a proof in the book "Topology and Geometry" by Glen Bredon. This is the proposition: ...
5
votes
2answers
73 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
1
vote
1answer
33 views

Definition of homotopy of slope fields

I can't come up with a correct definition of homotopic slope fields (on $\mathbb{R}^2$). Idea is clear - almost the same as vector field homotopy, but problem with defining slope as a function (case ...
0
votes
0answers
74 views

Precisely what is meant by “$\pi_1(M)$ is torsion”?

I am reading a paper where one of the conditions for a Theorem to hold is "the group $\pi_1(M)$ is torsion", where here $M$ is a compact differentiable manifold. What is meant by the first homotopy ...
2
votes
0answers
59 views

Homotopic maps between connected spaces inducing the same homomorphism between the fundamental groups

This is Problem 7-9 in Lee's Introduction to Topological Manifolds: Suppose $X$ and $Y$ are connected topological spaces, and the fundamental group of $Y$ is abelian. Show that if $F,G: X ...
3
votes
0answers
36 views

Is wedge sum for finite CW complexes cancellative in the homotopy category?

Let $X,Y,Z$ be finite pointed CW complexes. Is it possible that $X\vee Z$ and $Y\vee Z$ are homotopy equivalent, but $X$ and $Y$ are not? Remark 1: Without the finiteness assumption on $Z$, there are ...
1
vote
1answer
31 views

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ ...
9
votes
0answers
98 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
0
votes
2answers
42 views

A trivial fundamental group

I am reading fundamental groups from Munkres book. As stated in the definition, I understand a fundamental group relative to a base point $x_0$ includes all the loops based at point $x_0$. Later ...
6
votes
1answer
84 views

For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic

PROBLEM: For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic ,or even of degree 1 The following is my idea: First, choosing an arbitrary open coordinate ...
1
vote
0answers
22 views

How do the direct and inverse image sheaf functors interact with homotopy?

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for ...
6
votes
1answer
195 views

Map between $SO(n)$ is homotopic to the identity?

I'm given an exercise, in a differential geometry class, where I need to detemine wether or not the smooth map between manifolds: \begin{align} f \colon\ &SO(n) \rightarrow SO(n)\\ & A \mapsto ...
7
votes
1answer
164 views

Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$

Edit: As there are many comments and an answer already, I have left the original question below. I was unaware that there are different ways one could try to define $\mathsf{hTop}_{\bullet}$, 'the' ...
0
votes
0answers
41 views

Extend vector fields from several $S^1$ to $D^2$

Let's take a disk $D^2 \subset \mathbb{R}^2$ with $n$ holes ($n = 0, 1, ...$). In case $n = 0, 1$ it's clear how to extend any non-zero (i.e. with no singular points) vector field from $S^1$ to disk ...
1
vote
1answer
32 views

Prove (in the example) that being homotopic depends on the range of the Homotopy

Question: Define $F : [0,1]\times [0,1] \rightarrow X$ by $F(x, t) = (cos(\pi x), (1 - 2t) sin(\pi x))$. Take a straight-line homotopy between $F(x, 0)$ and $F(x, 1)$. Show that they are ...
7
votes
1answer
90 views

General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ...
1
vote
0answers
31 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
0
votes
2answers
41 views

Lift of $z^2$ map $S^1 \to S^1$

$\newcommand{\id}{\operatorname{id}}$It can be proved that identity map $\id: S^1 \to S^1$ does not lift to $\widetilde{\id} : S^1 \to \mathbb{R}$ such that $e^{\widetilde{\id(z)}i} = z$. The ...
0
votes
1answer
24 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
2
votes
1answer
30 views

explicit equivalent relation in the expression of the classifying space of a monoid

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The ...